secretuniversefandomcom-20200214-history
Zero term
A zero term is a product of the form \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*x or \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x*0 that is not equal to zero due to the requisite axioms that allow that to happen (usually the existence of additive inverses and cancellation) was discarded. Zero terms are very important in division by zero algebras as they control most of the addition structure of these algebras via the additive identity and their dominance properties. Zero terms are often idempotent, thus the study of division by zero algebras have close connection with the study of semigroups. Definition Zero term Given a non annihilating near-semiring \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}(S,*,+) with an element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x\in S and \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0\in S is an additive identity (i.e. there may be other one-sided identities), a left zero term is a nonzero expression \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*x^n , while a right zero term is a nonzero expression \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x^n*0 . It is a two-sided zero term (or just zero term) if it is both a left and right zero term, that is \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*0 . The order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n\in \mathbb{Z} of a zero term is the power of the element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x (which can be zero). Zero inverse Given a non annihilating near-semiring \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}(S,*,+) with an element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x\in S and \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0,1\in S is an additive identity and multiplicative identity respectively (i.e. there may be other one-sided identities), a left zero inverse is an element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x such that the right zero term \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x*0 equal to \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}1 , while a right zero inverse is an element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}x such that the left zero term \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*x equal to \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}1 . It is a two-sided zero inverse (or just the zero inverse) if it is both a left and right zero inverse. Like all inverses, the two sided inverse is unique. Properties To ease discussion, all near-semirings discussed below are non annihilating near-semirings (Otherwise zero terms cannot exist). Idempotence Theorem 1: All left zero terms are idempotent in a left distributive near-semiring, while all right zero terms are idempotent in a right distributive near-semiring. Proof: For all \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a\in S and suppose \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}S is distributive \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}a*0+a*0 & =a*(0+0) & \text{(Left distributivity)}\\ & = a*0 & \text{(One-sided additive identity)}\\ \end{align} Similarly for the right distributive case. Let \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}D\in\{L,R\} , this result can be compactly notated as \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a*0=_D a*0 Dominance Theorem 2a: Given a left additive identity \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0 , a left (resp. right) zero term of order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n+1 right absorbs a left (resp. right) zero term of order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n in a right (resp. left) distributive near-semiring. Similarly, given a right additive identity \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0 , a left (resp. right) zero term of order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n+1 left absorbs a left (resp. right) zero term of order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n in a right (resp. left) distributive near-semiring. Proof: \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}0*a^n+0*a^{n+1} & =(0+0*a)*a^n & \text{(Right distributivity)}\\ & = (0*a)*a^n & \text{(Left additive identity)}\\ & = 0*(a*a^n) & \text{(Associativity)}\\ & = 0*a^{n+1} \\ \end{align} and similarly for right zero terms and left distributivity. This is often stated as the zero term with respect to \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a of order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n+1 right dominates \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n . More compactly \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^{n+1} >_R 0*a^n . For the case of a right additive identity \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0 : \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}0*a^{n+1}+0*a^n & =(0*a+0)*a^n & \text{(Right distributivity)}\\ & = (0*a)*a^n & \text{(Right additive identity)}\\ & = 0*(a*a^n) & \text{(Associativity)}\\ & = 0*a^{n+1} \\ \end{align} and similarly for right zero terms and left distributivity. This is often stated as the zero term with respect to \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a of order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n+1 left dominates \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n . More compactly \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^{n+1} >_L 0*a^n . A zero term with respect to \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a of order \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n+1 dominates \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}n if it both left and right dominates with respect to \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a . That is, if both \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^{n+1} >_L 0*a^n and \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^{n+1} >_R 0*a^n . This is then stated simply as \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^{n+1} > 0*a^n . Therefore the type of the additive identity involved induces a natural D-partial ordering to the zero terms with respect to \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a of different orders, and in fact, powers of \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a if a corresponding zero inverse exists. Theorem 2a also made clear how the order of an element \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}1 hence the order of \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0 itself is determined by whether it is a left or right additive identity. Lemma 2.1: Suppose a two-sided multiplicative inverse of \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a exists, i.e. \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a^{-1} . Provided a compatible multiplicative identity exists, if \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a >_R 0 and \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^{-1} >_R 0 (i.e. 0 is a left additive identity), then \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0>_R 0*a^{-1} and \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0>_R 0*a respectively. Analogous results applies for right additive identities. Proof: \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}0*a^{-1}+0 & =0*a^{-1}+0*1 & \text{(Right multiplicative identity)}\\ & = 0*a^{-1}+0*(a*a^{-1}) & \text{(Multiplicative inverse of a)}\\ & = 0*a^{-1}+(0*a)*a^{-1} & \text{(Associativity)}\\ & = (0+(0*a))*a^{-1} & \text{(Right distributivity)}\\ & = (0*a)*a^{-1} & \text{(Left additive identity)}\\ & = 0*(a*a^{-1}) & \text{(Associativity)}\\ & = 0*1 & \text{(Multiplicative inverse of a)}\\ & = 0 & \text{(Right multiplicative identity)}\\ \end{align} \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}0*a+0 & =0*a+0*1 & \text{(Right multiplicative identity)}\\ & = 0*a+0*(a^{-1}*a) & \text{(Multiplicative inverse of a)}\\ & = 0*a+(0*a^{-1})*a & \text{(Associativity)}\\ & = (0+(0*a^{-1}))*a & \text{(Right distributivity)}\\ & = (0*a^{-1})*a & \text{(Left additive identity)}\\ & = 0*(a^{-1}*a) & \text{(Associativity)}\\ & = 0*1 & \text{(Multiplicative inverse of a)}\\ & = 0 & \text{(Right multiplicative identity)}\\ \end{align} and similarly for right zero terms with their corresponding identities. Lemma 2.1 also make it easy to see that if multiplicative inverses exists, in order to have the 'natural' looking D-partial ordered relation \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a > 0 > 0*a^{-1} and \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^{-1} > 0 > 0*a , at least two elements in the algebra will be absorbed by the additive identity, thus precluding the existence of an additive identity for this combination. Therefore, most near-semirings with zero terms often have a nonabelian addition structure if multiplicative inverses are present. In fact, any zero terms involved in this ordering will constrain the + structure such that the elements that are part of the ordering form a left or right null semigroup. This is why division by zero algebra have relatively trivial structures until associativity is broken. The combination above with the natural partial ordering, while cannot be a near-semiring due to the lack of additive identities, forms a ringnoid with unit and can be used to generate RPS magmas. Lemma 2.2: For all \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a,b \in S where \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a,b are not related by a D-partial order incompatible with the below assumption. If \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^n >_R 0*b then \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0 >_R 0*b*a^{-n} . Similarly for L cases. Proof: \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}0*b+0*a^n & =0*a^n & \text{(Right multiplicative identity)}\\ 0*b*a^{-n}+0*1 & =0*1 & \text{(Multiplicative inverse of a and associativity)}\\ 0*b*a^{-n}+0 & =0 & \text{(Right multiplicative identity)}\\ 0 & >_R 0*b*a^{-n} & \text{(Definition of D-partial order)}\\ \end{align} Similarly for L cases. Theorem 2b: For \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}m,n \in \mathbb{Z} . If \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}0*a^n >_R 0*a^m then \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} 0*a^{n-m} >_R 0 >_R 0*a^{m-n} . Similarly for L cases. Proof: \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white} \begin{align}0*a^n & >_R 0*a^m & \text{(Given)}\\ 0 & >_R 0*a^m*a^{-n} & \text{(Lemma 2.2 b= }a^m \text{)}\\ 0 & >_R 0*a^{m-n} & \text{(Definition of powers)}\\ 0*a^{n-m} & >_R 0 & \text{(Lemma 2.1)}\\ 0*a^{n-m} & >_R 0 >_R 0*a^{m-n} & \text{(Definition of D-partial order)}\\ \end{align} and similarly for L cases. When \definecolor{GGG}{RGB}{128,128,128}\pagecolor{GGG}\color{white}a is a zero inverse, Theorem 2b becomes the zero power laws. Usage Checking the D-partial ordering is crucial in the exploration of division by zero and division by zero divisor algebras due to the control they have on the addition structure. Theorem TBC Category:Division by zero